Relevant bibliographies by topics / Integral Equation Approach

Academic literature on the topic 'Integral Equation Approach'

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Published: 6 September 2023

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Journal articles on the topic "Integral Equation Approach"

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Haslinger, Jaroslav, C. C. Baniotopoulos, and Panagiotis D. Panagiotopoulos. "A boundary multivalued integral “equation” approach to the semipermeability problem." Applications of Mathematics 38, no. 1 (1993): 39–60. http://dx.doi.org/10.21136/am.1993.104533.

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Şenel, Ayşe Anapalı, Yalçın Öztürk, and Mustafa Gülsu. "New Numerical Approach for Solving Abel’s Integral Equations." Foundations of Computing and Decision Sciences 46, no. 3 (September 1, 2021): 255–71. http://dx.doi.org/10.2478/fcds-2021-0017.

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Abstract In this article, we present an efficient method for solving Abel’s integral equations. This important equation is consisting of an integral equation that is modeling many problems in literature. Our proposed method is based on first taking the truncated Taylor expansions of the solution function and fractional derivatives, then substituting their matrix forms into the equation. The main character behind this technique’s approach is that it reduces such problems to solving a system of algebraic equations, thus greatly simplifying the problem. Numerical examples are used to illustrate the preciseness and effectiveness of the proposed method. Figures and tables are demonstrated to solutions impress. Also, all numerical examples are solved with the aid of Maple.
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WHITFIELD, A. H., and N. MESSALI. "Integral-equation approach to system identification." International Journal of Control 45, no. 4 (April 1987): 1431–45. http://dx.doi.org/10.1080/00207178708933819.

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KNOBLES, D. P., S. A. STOTTS, R. A. KOCH, and T. UDAGAWA. "INTEGRAL EQUATION COUPLED MODE APPROACH APPLIED TO INTERNAL WAVE PROBLEMS." Journal of Computational Acoustics 09, no. 01 (March 2001): 149–67. http://dx.doi.org/10.1142/s0218396x01000449.

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A two-way coupled mode approach based on an integral equation formalism is applied to sound propagation through internal wave fields defined at the 1999 Shallow Water Acoustics Modeling Workshop. Solutions of the coupled equations are obtained using a powerful approach originally introduced in nuclear theory and also used to solve simple nonseparable problems in underwater acoustics. The basic integral equations are slightly modified to permit a Lanczos expansion to form a solution. The solution of the original set of integral equations is then easily recovered from the solution of the modified equations. Two important aspects of the integral equation method are revealed. First, the Lanczos expansion converges faster than a Born expansion of the original integral equations. Second, even when the Born expansion diverges due to strong mode coupling, the Lanczos expansion converges. It is shown that the internal wave problems examined are essentially one-way propagation problems because one observes good agreement between the coupled mode solutions and those provided by an energy-conserving parabolic equation algorithm. In the Workshop examples, at both 25 and 250 Hz, significantly greater coupling between modes occurs in the linear internal wave field case than the nonlinear soliton case.
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Wei, Tao, and Mingtian Xu. "An integral equation approach to the unsteady convection–diffusion equations." Applied Mathematics and Computation 274 (February 2016): 55–64. http://dx.doi.org/10.1016/j.amc.2015.10.084.

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Zozulya, V. V. "Divergent Integrals in Elastostatics: General Considerations." ISRN Applied Mathematics 2011 (August 2, 2011): 1–25. http://dx.doi.org/10.5402/2011/726402.

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This article considers weakly singular, singular, and hypersingular integrals, which arise when the boundary integral equation methods are used to solve problems in elastostatics. The main equations related to formulation of the boundary integral equation and the boundary element methods in 2D and 3D elastostatics are discussed in details. For their regularization, an approach based on the theory of distribution and the application of the Green theorem has been used. The expressions, which allow an easy calculation of the weakly singular, singular, and hypersingular integrals, have been constructed.
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Abdillah, Muhammad Taufik, Berlian Setiawaty, and Sugi Guritman. "The Solution of Generalization of the First and Second Kind of Abel’s Integral Equation." JTAM (Jurnal Teori dan Aplikasi Matematika) 7, no. 3 (July 17, 2023): 631. http://dx.doi.org/10.31764/jtam.v7i3.14193.

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Integral equations are equations in which the unknown function is found to be inside the integral sign. N. H. Abel used the integral equation to analyze the relationship between kinetic energy and potential energy in a falling object, expressed by two integral equations. This integral equation is called Abel's integral equation. Furthermore, these equations are developed to produce generalizations and further generalizations for each equation. This study aims to explain generalizations of the first and second kind of Abel’s integral equations, and to find solution for each equation. The method used to determine the solution of the equation is an analytical method, which includes Laplace transform, fractional calculus, and manipulation of equation. When the analytical approach cannot solve the equation, the solution will be determined by a numerical method, namely successive approximations. The results showed that the generalization of the first kind of Abel’s integral equation solution can be determined using the Laplace transform method, fractional calculus, and manipulation of equation. On the other hand, the generalization of the second kind of Abel’s integral equation solution is obtained from the Laplace transform method. Further generalization of the first kind of Abel’s integral equation solution can be obtained using manipulation of equation method. Further generalization of the second kind of Abel’s integral equation solution cannot be determined by analytical method, so a numerical method (successive approximations) is used.
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Chen, Jeng-Tzong, Chia-Chun Hsiao, and Shyue-Yuh Leu. "Null-Field Integral Equation Approach for Plate Problems With Circular Boundaries." Journal of Applied Mechanics 73, no. 4 (October 18, 2005): 679–93. http://dx.doi.org/10.1115/1.2165239.

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In this paper, a semi-analytical approach for circular plate problems with multiple circular holes is presented. Null-field integral equation is employed to solve the plate problems while the kernel functions in the null-field integral equation are expanded to degenerate kernels based on the separation of field and source points in the fundamental solution. The unknown boundary densities of the circular plates are expressed in terms of Fourier series. It is noted that all the improper integrals are transformed to series sum and are easily calculated when the degenerate kernels and Fourier series are used. By matching the boundary conditions at the collocation points, a linear algebraic system is obtained. After determining the unknown Fourier coefficients, the displacement, slope, normal moment, and effective shear force of the plate can be obtained by using the boundary integral equations. Finally, two numerical examples are proposed to demonstrate the validity of the present method and the results are compared with the available exact solution, the finite element solution using ABAQUS software and the data of Bird and Steele.
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Saadeh, Rania. "Applications of Double ARA Integral Transform." Computation 10, no. 12 (December 8, 2022): 216. http://dx.doi.org/10.3390/computation10120216.

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This paper describes our construction of a new double transform, which we call the double ARA transform (DARAT). Our novel double-integral transform can be used to solve partial differential equations and other problems. We discuss some fundamental characteristics of our approach, including existence, linearity, and several findings relating to partial derivatives and the double convolution theorem. DARAT can be used to precisely solve a variety of partial differential equations, including the heat equation, wave equation, telegraph equation, Klein–Gordon equation, and others, all of which are crucial for physical applications. Herein, we use DARAT to solve model integral equations to obtain exact solutions. We conclude that our novel method is easier to use than comparable transforms.
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Avdonin, S. A., B. P. Belinskiy, and John V. Matthews. "Inverse problem on the semi-axis: local approach." Tamkang Journal of Mathematics 42, no. 3 (August 24, 2011): 275–93. http://dx.doi.org/10.5556/j.tkjm.42.2011.916.

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We consider the problem of reconstruction of the potential for the wave equation on the semi-axis. We use the local versions of the Gelfand-Levitan and Krein equations, and the linear version of Simon's approach. For all methods, we reduce the problem of reconstruction to a second kind Fredholm integral equation, the kernel and the right-hand-side of which arise from an auxiliary second kind Volterra integral equation. A second-order accurate numerical method for the equations is described and implemented. Then several numerical examples verify that the algorithms can be used to reconstruct an unknown potential accurately. The practicality of each approach is briefly discussed. Accurate data preparation is described and implemented.
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Dissertations / Theses on the topic "Integral Equation Approach"

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Chapman, Geoffrey John Douglas. "A weakly singular integral equation approach for water wave problems." Thesis, University of Bristol, 2005. http://hdl.handle.net/1983/54f56a00-8496-4990-8410-d2c677839095.

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Messali, Nouari. "An integral equation approach to continuous system identification and model reduction." Thesis, Loughborough University, 1988. https://dspace.lboro.ac.uk/2134/33193.

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An integral equation description for linear systems is developed and used as the basis for the development of various system identification, model reduction and order determination methods. The system integral equation is utilized in the problem of parameter identification in continuous linear single-input single-output, multi-input multi-output and linear in parameters nonlinear systems. The approach is developed in the time domain where the effect of non-zero initial conditions and additive disturbances occurs naturally. Parameter estimates are deduced using several weighted residual concepts which have previously been used to produce approximate solutions to differential equations.
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Chamberlain, Peter George. "Wave propagation on water of uneven depth : an integral equation approach." Thesis, University of Reading, 1991. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.304583.

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Menotti, Enrico. "Time-dependent and three-dimensional phenomena in free-electron laser amplifiers within the integral-equation approach." Doctoral thesis, Università degli studi di Trieste, 2011. http://hdl.handle.net/10077/4485.

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O'Donoghue, Padraic Eimear. "Boundary integral equation approach to nonlinear response control of large space structures : alternating technique applied to multiple flaws in three dimensional bodies." Diss., Georgia Institute of Technology, 1985. http://hdl.handle.net/1853/20685.

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Kulkarni, Shashank D. "Development and validation of a Method of Moments approach for modeling planar antenna structures." Worcester, Mass. : Worcester Polytechnic Institute, 2007. http://www.wpi.edu/Pubs/ETD/Available/etd-042007-151741/.

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Dissertation (Ph.D.)--Worcester Polytechnic Institute.
Keywords: patch antennas; volume integral equation (VIE); method of moments (MoM); low order basis functions; convergence. Includes bibliographical references (leaves 169-186 ).
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Pipis, Konstantinos. "Eddy-current testing modeling of axisymmetric pieces with discontinuities along the axis by means of an integral equation approach." Thesis, Université Paris-Saclay (ComUE), 2015. http://www.theses.fr/2015SACLS176/document.

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Le contrôle non destructif (CND) de pièces pour des applications dans l'industrie a mené au besoin de modèles rapides et précises. Tels modèles servent au développement des méthodes d'inspection, à l'optimisation des capteurs utilisés aux essais, à l'évaluation des courbes de Probabilité de Detection (POD) ainsi qu'à la caractérisation de défauts. Cette thèse se focalise au CND par Courants de Foucault (CF) de pièces cylindriques avec des discontinuités selon z et contenant un défaut fin. Un modèle pour l'inspection de telles pièces a été développé afin de traiter des applications comme l'inspection des pièces alésées trouvées en aéronautique et des tubes des générateurs de vapeur utilisés dans l'industrie nucléaire. Ce modèle est basé sur une formulation d'équation intégrale. Plus précisément, la variation de l'impédance du capteur, dit signal CF, est calculée à partir d'une équation intégrale sur la surface du défaut. La formulation suivie est basée sur la méthode d'intégration surfacique (SIM). Cette formulation nécessite, d'un côté, le calcul du champ électrique en absence du défaut et, de l'autre côté, l'expression d'une fonction de Green qui correspond à la géométrie de la pièce sans défaut. Les deux problèmes électromagnétiques sont résolus en utilisant la méthode Truncation Region Eigenfunction Expansion (TREE). La méthode TREE est un outil performant pour la résolution des problèmes électromagnétiques qui prend en compte la décroissance rapide de l'intensité du champ afin de tronquer le domaine d'intérêt à une distance, où le champ est négligeable.Le modèle est validé en comparant le signal CF calculé avec des résultats obtenues par une approche combinant la méthode d'intégration volumique (VIM) et SIM, dite l'approche VIM-SIM (implémentée dans la plateforme CIVA) ainsi qu'avec le modèle d'éléments finis (FEM). Nous avons traité trois configurations différentes : un demi-espace conducteur alésé avec un défaut fin, une plaque conductrice avec un alésage et un défaut, et un tube semi-infini avec un défaut fin à la proximité de son bord. La comparaison des résultats montre un très bon accord entre les trois modèles. Le temps de calcul avec le modèle SIM est considérablement inférieur aux temps de calcul des autres modèles. En outre, le modèle SIM donne la possibilité d'effectuer le balayage du capteur dans le tube ou l'alésage dans le cas des pièces alésées
Nondestructive Testing (NDT) of parts for industrial applications such as in nuclear and aeronautical industry has led to the need for fast and precise models. Such models are useful for the development of the inspection methods, the optimisation of probes, the evaluation of the Probability of Detection (POD) curves or for the flaw characterisation.This PhD thesis focuses on the eddy-current NDT of layered cylindrical pieces with discontinuities in the z direction and containing a narrow crack. A model for the inspection of such pieces is developed in order to be applied on the inspection of fastener holes met in aeronautics and of steam generator tubes in nuclear sector.The model is based on an integral equation formalism. More precisely, for the calculation of the impedance change one needs to solve an integral equation over the surface of the narrow crack, which is represented by a surface electric dipole distribution. This is the method known as surface integration method (SIM). This formulation requires, on the one hand, the calculation of the electric field in the absence of the flaw, the so-called primary field, and, on the other hand, the Green's function expression corresponding to the geometry of the flawless piece. Both electromagnetic problems are solved by means of the Truncation Region Eigenfunction Expansion (TREE) method. The TREE method is a powerful tool for the solution of electromagnetic problems which uses the rapid decrease of the field in order to truncate the region of interest at a distance where the field is negligible.The model is validated by comparing the results of the coil impedance variation with those obtained by an approach that combines the volume integral method (VIM) with SIM, known as VIM-SIM method, implemented in the commercial software CIVA and the finite element method (FEM) implementation in COMSOL software. Three different configurations have treated. The more general geometry of a conducting half-space with a borehole, a conducting plate with a borehole and a crack and a conducting semi-infinite tube with a crack near the edge. The results of the three models show good agreement between them. The computational time of the SIM model is significantly lower compared to previous models. Furthermore, another advantage of the SIM model is that it provides the possibility of a scan inside the borehole
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Murakami, Shota. "Theoretical Prediction of Changes in Protein Structural Stability upon Cosolvent or Salt Addition and Amino-acid Mutation." 京都大学 (Kyoto University), 2017. http://hdl.handle.net/2433/225706.

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Michelis, Katina. "A sequential eigenfunction expansion approach for certain nonlinear integral equations." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1998. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape11/PQDD_0006/MQ44221.pdf.

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Michelis, Katina. "A sequential eigenfunction expansion approach for certain nonlinear integral equations /." Thesis, McGill University, 1997. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=20588.

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In the numerical solution of nonlinear integral equations, classic finite difference and series methods lead to systems of nonlinear algebraic or transcendental equations which are solved by iterative schemes such as the Newton method. The present work develops a sequential eigenfunction expansion for the numerical solution of certain nonlinear integral equations. The nonlinear term provides constraints for the amplitudes of the eigenfunctions and a subsequent iteration is used to refine these coefficients. A comparative study of the present method with the Broyden method is conducted. It is shown that the expansion procedure provides an early indication about the multiplicity of solutions which is not present when using the classic methods of solution. Numerical examples are presented which demonstrate the robustness of the expansion method in determining multiple solutions.
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Books on the topic "Integral Equation Approach"

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Słobodzian, Piotr M. Electromagnetic analysis of shielded microwave structures: The surface integral equation approach. Wrocław: Oficyna Wydawnicza Politechniki Wrocławskiej, 2007.

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Sin-Chung, Chang, and United States. National Aeronautics and Space Administration., eds. The Space-time solution element method-a new numerical approach for the Navier-Stokes equations. [Washington, DC]: National Aeronautics and Space Administration, 1995.

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Sin-Chung, Chang, and United States. National Aeronautics and Space Administration., eds. The Space-time solution element method-a new numerical approach for the Navier-Stokes equations. [Washington, DC]: National Aeronautics and Space Administration, 1995.

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Bogdanov, L. V. Analytic-Bilinear Approach to Integrable Hierarchies. Dordrecht: Springer Netherlands, 1999.

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Assing, Sigurd. Continuous strong Markov processes in dimension one: A stochastic calculus approach. Berlin: Springer, 1998.

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Stochastic integration and differential equations: A new approach. 2nd ed. Berlin: Springer-Verlag, 1992.

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The computational complexity of differential and integral equations: An information-based approach. Oxford [England]: Oxford University Press, 1991.

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Protter, Philip E. Stochastic integration and differential equations: A new approach. Berlin: Springer-Verlag, 1990.

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1950-, Panagiotopoulos P. D., ed. The boundary integral approach to static and dynamic contact problems: Equality and inequality methods. Basel: Birkhäuser, 1992.

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Spain) UIMP-RSME Lluis Santaló Summer (2012 Santander. Recent advances in real complexity and computation: UIMP-RSME Lluis A. Santaló Summer School, Recent advances in real complexity and computation, July 16-20, 2012, Universidad Internacional Menéndez Pelayo, Santander, Spain. Edited by Montaña, Jose Luis, 1961- editor of compilation and Pardo, L. M. (Luis M.), editor of compilation. Providence, Rhode Island: American Mathematical Society, 2013.

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Book chapters on the topic "Integral Equation Approach"

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Tokovyy, Yuriy V., Bohdan M. Kalynyak, and Chien-Ching Ma. "Nonhomogeneous Solids: Integral Equation Approach." In Encyclopedia of Thermal Stresses, 3350–56. Dordrecht: Springer Netherlands, 2014. http://dx.doi.org/10.1007/978-94-007-2739-7_615.

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Fujiwara, Daisuke. "Feynman Path Integral and Schrödinger Equation." In Rigorous Time Slicing Approach to Feynman Path Integrals, 137–86. Tokyo: Springer Japan, 2017. http://dx.doi.org/10.1007/978-4-431-56553-6_6.

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Tanaka, M. "New Integral Equation Approach to Viscoelastic Problems." In Computational Aspects, 25–35. Berlin, Heidelberg: Springer Berlin Heidelberg, 1987. http://dx.doi.org/10.1007/978-3-642-82663-4_2.

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Sandhas, W. "Integral Equation Approach to Few-Body Collision Problems." In Theoretical and Experimental Investigations of Hadronic Few-Body Systems, 64–78. Vienna: Springer Vienna, 1986. http://dx.doi.org/10.1007/978-3-7091-8897-2_7.

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Tanaka, Michihiko, and Kanji Fujii. "Boundary Integral Equation Approach to the Classical Theory of Elasticity." In Computational Mechanics ’95, 2726–31. Berlin, Heidelberg: Springer Berlin Heidelberg, 1995. http://dx.doi.org/10.1007/978-3-642-79654-8_452.

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Deville, Michel O. "Boundary Layer." In An Introduction to the Mechanics of Incompressible Fluids, 175–95. Cham: Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-031-04683-4_7.

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AbstractThe Prandtl’s equations for laminar boundary layer are obtained via dimensional analysis. The case of the flat plate is treated as a suitable example for the development of the boundary layer on a simple geometry. Various thicknesses are introduced. The integration of Prandtl’s equation across the boundary layer produces the von Kármán integral equation which allows the elaboration of the approximate von Kármán-Pohlhausen method where the velocity profile is given as a polynomial. The use of a third degree polynomial for the flat plate demonstrates the feasibility of the approach.
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Yagdjian, Karen. "Integral Transform Approach to Solving Klein–Gordon Equation with Variable Coefficients." In Theory, Numerics and Applications of Hyperbolic Problems II, 655–64. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-91548-7_49.

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Chao, J., N. Nakagawa, D. Raulerson, and J. Moulder. "A General Boundary Integral Equation Approach to Eddy Current Crack Modeling." In Review of Progress in Quantitative Nondestructive Evaluation, 279–85. Boston, MA: Springer US, 1997. http://dx.doi.org/10.1007/978-1-4615-5947-4_36.

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Sloss, J. M., J. C. Bruch, S. Adali, and I. S. Sadek. "An Integral Equation Approach for Velocity Feedback Control Using Piezoelectric Patches." In IUTAM Symposium on Smart Structures and Structronic Systems, 331–38. Dordrecht: Springer Netherlands, 2001. http://dx.doi.org/10.1007/978-94-010-0724-5_41.

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Cosgun, Tahir, Murat Sari, and Hande Uslu. "A New Approach for the Solution of the Generalized Abel Integral Equation." In Nonlinear Systems and Complexity, 145–51. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-37141-8_8.

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Conference papers on the topic "Integral Equation Approach"

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Moroney, D. "An integral equation approach to UHF coverage estimation." In Ninth International Conference on Antennas and Propagation (ICAP). IEE, 1995. http://dx.doi.org/10.1049/cp:19950452.

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Zhang, J., and M. S. Tong. "Integral equation approach for analyzing electromagnetic radiation of elastic objects." In 2013 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting. IEEE, 2013. http://dx.doi.org/10.1109/aps.2013.6711321.

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Janaswamy, R. "A Fredholm integral equation approach to propagation over irregular terrain." In IEEE Antennas and Propagation Society International Symposium 1992 Digest. IEEE, 1992. http://dx.doi.org/10.1109/aps.1992.221696.

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Ngobigha, Felix O., and David H. O. Bebbington. "Electromagnetic waves scattering by dielectric ellipsoids applying integral equation approach." In 2014 USNC-URSI Radio Science Meeting (Joint with AP-S Symposium). IEEE, 2014. http://dx.doi.org/10.1109/usnc-ursi.2014.6955610.

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Sandu, Titus, George Boldeiu, and Rodica Plugaru. "Charging and capacitance of conductors by the integral equation approach." In 2013 International Semiconductor Conference (CAS 2013). IEEE, 2013. http://dx.doi.org/10.1109/smicnd.2013.6688679.

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Narahari Achar, B. N., and John W. Hanneken. "Response Dynamics in the Continuum Limit of the Lattice Dynamical Theory of Viscoelasticity (Fractional Calculus Approach)." In ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2009. http://dx.doi.org/10.1115/detc2009-86218.

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A fractional diffusion-wave equation is derived in the continuum limit of the lattice dynamical equations of motion of a chain of coupled fractional oscillators obtained from the integral equations of motion of a linear chain of simple harmonic oscillators by generalization of the ordinary integrals into ones involving fractional integrals. The set of integral equations of motion pertaining to the chain of coupled fractional oscillators in the continuum limit is solved by using Laplace transforms. The response of the system to impulse and sinusoidal forcing is studied. Numerical applications are discussed with particular reference to energy flow and dissipation.
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Chen, X. Z., J. Zhang, J. H. Zhou, X. F. Yin, and M. S. Tong. "A meshless scheme for reconstructing dielectric objects by integral equation approach." In 2013 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting. IEEE, 2013. http://dx.doi.org/10.1109/aps.2013.6710842.

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Leon, L. J., and F. A. Roberge. "Excitation in a cylinder of cardiac membrane: an integral equation approach." In Proceedings of the Annual International Conference of the IEEE Engineering in Medicine and Biology Society. IEEE, 1988. http://dx.doi.org/10.1109/iembs.1988.94471.

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Petek, M., J. Rivero, J. A. Vasquez-Tobon, G. Valerio, O. Quevedo-Teruel, and F. Vipiana. "Efficient Integral Equation Approach for the Modelling of Glide-Symmetric Structures." In 2023 17th European Conference on Antennas and Propagation (EuCAP). IEEE, 2023. http://dx.doi.org/10.23919/eucap57121.2023.10133317.

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KANDIL, O., and E. YATES, JR. "Computation of transonic vortex flows past delta wings Integral equation approach." In 18th Fluid Dynamics and Plasmadynamics and Lasers Conference. Reston, Virigina: American Institute of Aeronautics and Astronautics, 1985. http://dx.doi.org/10.2514/6.1985-1582.

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Reports on the topic "Integral Equation Approach"

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Weile, Daniel S. A Time Domain Integral Equation Approach to Electromagnetic Interference Simulation. Fort Belvoir, VA: Defense Technical Information Center, June 2007. http://dx.doi.org/10.21236/ada471318.

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Guderley, Karl G. A Semi-Analytical Approach to the Integral Equation (in Terms of the Acceleration Potential) for the Linearized Subsonic, Oscillatory Flow Over an Airfoil. Fort Belvoir, VA: Defense Technical Information Center, March 1986. http://dx.doi.org/10.21236/ada167313.

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3

Ostashev, Vladimir, Michael Muhlestein, and D. Wilson. Extra-wide-angle parabolic equations in motionless and moving media. Engineer Research and Development Center (U.S.), September 2021. http://dx.doi.org/10.21079/11681/42043.

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Abstract:
Wide-angle parabolic equations (WAPEs) play an important role in physics. They are derived by an expansion of a square-root pseudo-differential operator in one-way wave equations, and then solved by finite-difference techniques. In the present paper, a different approach is suggested. The starting point is an extra-wide-angle parabolic equation (EWAPE) valid for small variations of the refractive index of a medium. This equation is written in an integral form, solved by a perturbation technique, and transformed to the spectral domain. The resulting split-step spectral algorithm for the EWAPE accounts for the propagation angles up to 90° with respect to the nominal direction. This EWAPE is also generalized to large variations in the refractive index. It is shown that WAPEs known in the literature are particular cases of the two EWAPEs. This provides an alternative derivation of the WAPEs, enables a better understanding of the underlying physics and ranges of their applicability, and opens an opportunity for innovative algorithms. Sound propagation in both motionless and moving media is considered. The split-step spectral algorithm is particularly useful in the latter case since complicated partial derivatives of the sound pressure and medium velocity reduce to wave vectors (essentially, propagation angles) in the spectral domain.
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